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A216134 Numbers n such that T_n and 2*T_n + 1 are triangular. 16
0, 1, 4, 9, 26, 55, 154, 323, 900, 1885, 5248, 10989, 30590, 64051, 178294, 373319, 1039176, 2175865, 6056764, 12681873, 35301410, 73915375, 205751698, 430810379, 1199208780, 2510946901, 6989500984, 14634871029, 40737797126, 85298279275, 237437281774 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Entries in this sequence are the indices of the Sophie Germain triangular numbers A124174.

This sequence is intimately associated with the Pell numbers (A000129) and Half-companion Pell numbers (A001333), as can be seen in the formula section below and by the following identities:

sqrt(2) = lim_{k->infinity} ((a(2k+1) + a(2k) + 1)/2)/(a(2k+1) - a(2k)); e.g. ((85298279275 + 40737797126)/2)/(85298279275 - 40737797126) = 1.414213562361... for k = 14. Same as lim_{k->infinity} A001333(2k + 1)/A000129(2k + 1).

1 + (sqrt 2) = lim_{k->infinity} (a(2k + 1) - a(2k))/(a(2k + 1) - 2*a(2k) +  a(2k - 1)); e.g., (85298279275 - 40737797126)/(85298279275 - 2*40737797126 + 14634871029) = 2.414213562373... for k = 14. Same as lim_{k->infinity} A000129(2k + 1)/A000129(2k).

1 + 1/(sqrt 2) = lim_{k->infinity} (a(2k+1) - a(2k))/(a(2k) - a(2k - 1)); e.g., (85298279275 - 40737797126)/(40737797126 - 14634871029) = 1.707106781186... for k = 14. Same as lim_{k->infinity} A000129(2k + 1)/A001333(2k).

Numbers n such that 2*triangular(n) + 1 is a triangular number. Equivalently, numbers n such that n^2 + n + 1 is a triangular number. - Alex Ratushnyak, Apr 18 2013

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Wikipedia, Pell numbers

Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

FORMULA

G.f.: x*(1+3*x-x^2-x^3)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2)). - R. J. Mathar, Sep 08 2012

If, for k in N, ...

z = (k + (k mod 2) + 2)/2 - (k mod 2),

and

y = A006452(k) = (((3 + sqrt(8))^z - (3 - sqrt(8))^z) - 2*((3 + sqrt(8))^(z-2) - (3 - sqrt(8))^(z-2))) / ((4 - 2(-1)^k)*sqrt(8))

Then,

n = y + ((sqrt(8y^2 - 7) - 1)/2 - (1 - sign(k))).

Additionally, for all terms

n = (sqrt(8*A124174(k) + 1) - 1)/2

n = A006451(k+1) - A006452(k+2)

n = A124124(k) - A006452(k)

n = (sqrt(4*((A124124(k)*(A124124(k) + 1))/2) - 3) - 1)/2

... for all terms except 0

n = A006451(k) + A006452(k+1)

n = SUM A079496(k).

a(n) = (2*A000129(n) + (-1)^n*(A000129(2*floor(n/2) - 1) - (-1)^n)/2); A000129 gives the Pell numbers. - Raphie Frank, Jan 04 2013

From Raphie Frank, Jan 04 2013: (Start)

A124174(n) = a(n)*(a(n) + 1)/2; Sophie Germain triangular numbers.

A079496(n) = a(n + 1) - a(n).

A000129(2n) = a(2n) - 2*a(2n - 1) +  a(2n - 2); even-indexed Pell numbers.

A000129(2n) = a(2n + 1) - 2*a(2n) +  a(2n - 1); even-indexed Pell numbers.

A000129(2n + 1) = a(2n + 1) - a(2n); odd-indexed Pell numbers.

A001333(2n) = a(2n) - a(2n - 1); even-indexed Half-companion Pell numbers.

A001333(2n + 1) = (a(2n + 1) + a(2n) + 1)/2; odd-indexed Half-companion Pell numbers.

A006451(n + 1) = (a(n + 2) + a(n))/2.

A006452(n + 2) = (a(n + 2) - a(n))/2.

A124124(n + 2) = (a(n + 2) + a(n))/2 + (a(n + 2) - a(n)).

(End)

a(n + 2) = sqrt(8*a(n)^2 + 8*a(n) + 9) + 3*a(n) + 1; a(0) = 0, a(1) = 1. - Raphie Frank, Feb 02 2013

a(n) = (3/8 + sqrt(2)/4)*(1 + sqrt(2))^n + (-1/8 - sqrt(2)/8)*(-1 + sqrt(2))^n + (3/8 - sqrt(2)/4)*(1 - sqrt(2))^n + (-1/8 + sqrt(2)/8)*(-1 - sqrt(2))^n - 1/2. - Robert Israel, Aug 13 2014

E.g.f.: (1/4)*(-2*cosh(x) - 2*sinh(x) + 2*cosh(sqrt(2)*x)*(cosh(x) + 2*sinh(x)) + sqrt(2)*(cosh(x) + 3*sinh(x))*sinh(sqrt(2)*x)). - Stefano Spezia, Dec 10 2019

EXAMPLE

11 + ((sqrt(8*11^2 - 7) - 1)/2 - (1 - sign(4)))  = 11 + (15 - 0) = 26.

23 + ((sqrt(8*23^2 - 7) - 1)/2 - (1 - sign(5)))  = 23 + (32 - 0) = 55.

64 + ((sqrt(8*64^2 - 7) - 1)/2 - (1 - sign(6)))  = 64 + (90 - 0) = 154.

a(8) = sqrt(8*a(6)^2 + 8*a(6) + 9) + 3*a(6) + 1 = sqrt(8*154^2 + 8*154 + 9) + 3*154 + 1 = 900. - Raphie Frank, Feb 02 2013

MATHEMATICA

LinearRecurrence[{1, 6, -6, -1, 1}, {0, 1, 4, 9, 26}, 40] (* T. D. Noe, Sep 03 2012 *)

PROG

(PARI) Vec( x*(1+3*x-x^2-x^3)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2)) + O(x^66) ) \\ Joerg Arndt, Aug 13 2014

(PARI) isok(n) = ispolygonal(n*(n+1) + 1, 3); \\ Michel Marcus, Aug 13 2014

CROSSREFS

Cf. A124174, A000129, A001333, A006451, A006452, A124124, A079496.

Cf. A069017 (triangular numbers of the form k^2 + k + 1).

Sequence in context: A090117 A329125 A020181 * A226908 A328657 A113682

Adjacent sequences:  A216131 A216132 A216133 * A216135 A216136 A216137

KEYWORD

nonn,easy

AUTHOR

Raphie Frank, Sep 01 2012

EXTENSIONS

Restructured comments section, with additions, to better show relationship of this sequence to the Pell numbers and Half-companion Pell numbers. Also added appropriate cross-references. - Raphie Frank, Jan 04 2013

STATUS

approved

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)